1 Introduction

# Micmac Manual
###### \[in package MICMAC\]
## micmac ASDF System Details
- Version: 0.0.2
- Description: Micmac is mainly a library of graph search algorithms
such as alpha-beta, UCT and beam search, but it also has some MCMC
and other slightly unrelated stuff.
- Licence: MIT, see COPYING.
- Author: Gábor Melis
- Mailto: [mega@retes.hu](mailto:mega@retes.hu)
- Homepage: [http://quotenil.com](http://quotenil.com)
## Introduction
### Overview
MICMAC is a Common Lisp library by [Gábor
Melis](http://quotenil.com) focusing on [graph
search](http://en.wikipedia.org/wiki/Graph_traversal) algorithms.
### Links
Here is the [official
repository](https://github.com/melisgl/micmac) and the [HTML
documentation](http://melisgl.github.io/mgl-pax-world/micmac-manual.html)
for the latest version.
## Graph Search
- [function] ALPHA-BETA STATE &KEY (DEPTH 0) ALPHA BETA CALL-WITH-ACTION MAYBE-EVALUATE-STATE LIST-ACTIONS RECORD-BEST
Alpha-beta pruning for two player, zero-sum maximax (like minimax
but both players maximize and the score is negated when passed
between depths). Return the score of the game STATE from the point
of view of the player to move at DEPTH and as the second value the
list of actions of the principal variant.
CALL-WITH-ACTION is a function of (STATE DEPTH ACTION FN). It
carries out ACTION (returned by LIST-ACTIONS or NIL) to get the
state corresponding to DEPTH and calls FN with that state. It may
destructively modify STATE provided it undoes the damage after FN
returns. CALL-WITH-ACTION is called with NIL as ACTION for the root
of the tree, in this case STATE need not be changed. FN returns the
same kinds of values as ALPHA-BETA. They may be useful for logging.
MAYBE-EVALUATE-STATE is a function of (STATE DEPTH). If STATE at
DEPTH is a terminal node then it returns the score from the point of
view of the player to move and as the second value a list of actions
that lead from STATE to the position that was evaluated. The list of
actions is typically empty. If we are not at a terminal node then
MAYBE-EVALUATE-STATE returns NIL.
LIST-ACTIONS is a function of (STATE DEPTH) and returns a non-empty
list of legal candidate moves for non-terminal nodes. Actions are
tried in the order LIST-ACTIONS returns them: stronger moves
CALL-WITH-ACTION, MAYBE-EVALUATE-STATE and LIST-ACTIONS are
mandatory.
RECORD-BEST, if non-NIL, is a function of (DEPTH SCORE ACTIONS). It
is called when at DEPTH a new best action is found. ACTIONS is a
list of all the actions in the principle variant corresonding to the
newly found best score. RECORD-BEST is useful for graceful
degradation in case of timeout.
ALPHA and BETA are typically NIL (equivalent to -infinity,
+infinity) but any real number is allowed if the range of scores can
be boxed.
See `test/test-alpha-beta.lisp` for an example.
- [function] BEAM-SEARCH START-NODES &KEY MAX-DEPTH (N-SOLUTIONS 1) (BEAM-WIDTH (LENGTH START-NODES)) EXPAND-NODE-FN EXPAND-BEAM-FN SCORE-FN UPPER-BOUND-FN SOLUTIONP-FN (FINISHEDP-FN SOLUTIONP-FN)
In a graph, search for nodes that with the best scores with [beam
search](http://en.wikipedia.org/wiki/Beam_search). That is, starting
from START-NODES perform a breadth-first search but at each depth
only keep BEAM-WIDTH number of nodes with the best scores. Keep the
best N-SOLUTIONS (at most) complete solutions. Discard nodes known
to be unable to get into the best N-SOLUTIONS (due to
UPPER-BOUND-FN). Finally, return the solutions and the active
nodes (the *beam*) as adjustable arrays sorted by score in
descending order.
START-NODES (a sequence of elements of arbitrary type). SCORE-FN,
UPPER-BOUND-FN, SOLUTIONP-FN, FINISHEDP-FN are all functions of one
argument: the node. SOLUTIONP-FN checks whether a node represents a
complete solution (i.e. some goal is reached). SCORE-FN returns a
real number that's to be maximized, it's only called for node for
which SOLUTIONP-FN returned true. UPPER-BOUND-FN (if not NIL)
returns a real number that equal or greater than the score of all
solutions reachable from that node. FINISHEDP-FN returns true iff
there is nowhere to go from the node.
EXPAND-NODE-FN is also a function of a single node argument. It
returns a sequence of nodes to 'one step away' from its argument
node. EXPAND-BEAM-FN is similar, but it takes a vector of nodes and
returns all nodes one step away from any of them. It's enough
provide either EXPAND-NODE-FN or EXPAND-BEAM-FN. The purpose of
EXPAND-BEAM-FN. is to allow more efficient, batch-like operations.
See `test/test-beam-search.lisp` for an example.
- [function] PARALLEL-BEAM-SEARCH START-NODE-SEQS &KEY MAX-DEPTH (N-SOLUTIONS 1) BEAM-WIDTH EXPAND-NODE-FN EXPAND-BEAMS-FN SCORE-FN UPPER-BOUND-FN SOLUTIONP-FN (FINISHEDP-FN SOLUTIONP-FN)
This is very much like BEAM-SEARCH except it solves a number of
instances of the same search problem starting from different sets of
nodes. The sole purpose of PARALLEL-BEAM-SEARCH is to amortize the
cost EXPAND-BEAM-FN if possible.
EXPAND-BEAMS-FN is called with sequence of beams (i.e. it's a
sequence of sequence of nodes) and it must return another sequence
of sequences of nodes. Each element of the returned sequence is the
reachable nodes of the nodes in the corresponding element of its
argument sequence.
PARALLEL-BEAM-SEARCH returns a sequence of solutions sequences, and
a sequence of active node sequences.
See `test/test-beam-search.lisp` for an example.
### UCT
###### \[in package MICMAC.UCT\]
UCT Monte Carlo tree search. This is what makes current Go programs
tick. And Hex programs as well, for that matter. This is a cleanup
and generalization of code originally created in course of the
Google AI Challenge 2010.
For now, the documentation is just a reference. See
`test/test-uct.lisp` for an example.
- [class] UCT-NODE
A node in the UCT tree. Roughly translates to a
state in the search space. Note that the state itself is not stored
explicity, but it can be recovered by \`replaying' the actions from
the starting state or by customizing MAKE-UCT-NODE.
- [reader] DEPTH UCT-NODE (:DEPTH = 0)
- [accessor] EDGES UCT-NODE
Outgoing edges.
- [accessor] AVERAGE-REWARD UCT-NODE (:AVERAGE-REWARD = 0)
Average reward over random playouts started from
below this node. See UPDATE-UCT-STATISTICS and REWARD.
- [class] UCT-EDGE
An edge in the UCT tree. Represents an action taken
from a state. The value of an action is the value of its target
state which is not quite as generic as it could be; feel free to
specialize AVERAGE-REWARD for the edges if that's not the case.
- [reader] ACTION UCT-EDGE (:ACTION)
The action represented by the edge.
- [accessor] FROM-NODE UCT-EDGE (:FROM-NODE)
The node this edge starts from.
- [accessor] TO-NODE UCT-EDGE (= NIL)
The node this edge points to if the edge has been
visited or NIL.
- [function] VISITED-EDGES NODE
- [function] UNVISITED-EDGES NODE
- [generic-function] EDGE-SCORE NODE EDGE EXPLORATION-BIAS
- [generic-function] SELECT-EDGE NODE EXPLORATION-BIAS
Choose an action to take from a state, in other
words an edge to follow from NODE in the tree. The default
implementation chooses randomly from the yet unvisited edges or if
there is none moves down the edge with the maximum EDGE-SCORE. If
you are thinking of customizing this, for example to make it choose
the minimum at odd depths, the you may want to consider specializing
REWARD or UPDATE-UCT-STATISTICS instead.
- [generic-function] OUTCOME->REWARD NODE OUTCOME
Compute the reward for a node in the tree from
OUTCOME that is the result of a playout. This is called by the
default implementation of UPDATE-UCT-STATISTICS. This is where one
typically negates depending on the parity of DEPTH in two player
games.
- [generic-function] UPDATE-UCT-STATISTICS ROOT PATH OUTCOME
Increment the number of visits and update the
average reward in nodes and edges of PATH. By default, edges simply
get their visit counter incremented while nodes also get an update
to AVERAGE-REWARD based on what OUTCOME->REWARD returns.
- [generic-function] MAKE-UCT-NODE PARENT EDGE PARENT-STATE
Create a node representing the state of that EDGE
leads to from PARENT. Specialize this if you want to keep track of
the state which is not done by default as it can be expensive,
especially in light of TAKE-ACTION mutating it. The default
implementation simply creates an instance of the class of PARENT so
that one can start from a subclass of UCT-NODE and be sure that that
class is going to be used for nodes below it.
- [generic-function] STATE NODE PARENT EDGE PARENT-STATE
Return the state that corresponds to NODE. This is
not a straightforward accessor unless NODE is customized to store
it. The rest of the parameters are provided so that one can
reconstruct the state by taking the action of EDGE in the
PARENT-STATE of PARENT. It's okay to destroy PARENT-STATE in the
process as long as it's not stored elsewhere. This function must be
customized.
- [generic-function] LIST-EDGES NODE STATE
Return a list of edges representing the possible
actions from NODE with STATE. This function must be customized.
- [generic-function] PLAY-OUT NODE STATE REVERSE-PATH
Play a random game from NODE with STATE and return
the outcome that's fed into UPDATE-UCT-STATISTICS. The way the
random game is played is referred to as \`default policy' and that's
what makes or breaks UCT search. This function must be
customized.
- [function] UCT &KEY ROOT FRESH-ROOT-STATE EXPLORATION-BIAS MAX-N-PLAYOUTS
Starting from the ROOT node search the tree expanding it one node
for each playout. Finally return the mutated ROOT. ROOT may be the
root node of any tree, need not be a single node with no edges.
FRESH-ROOT-STATE is a function that returns a fresh state
corresponding to ROOT. This state will be destroyed unless special
care is taken in STATE.
## Metropolis Hastings
###### \[in package MICMAC.METROPOLIS-HASTINGS\]
Generic interface for the Metropolis-Hastings algorithm, also
Metropolis Coupled MCMC.
References:
- http://en.wikipedia.org/wiki/Metropolis–Hastings\_algorithm
- Markov Chain Monte Carlo and Gibbs Sampling
Lecture Notes for EEB 581, version 26 April 2004 c B. Walsh 2004
http://web.mit.edu/~wingated/www/introductions/mcmc-gibbs-intro.pdf
- Geyer, C.J. (1991) Markov chain Monte Carlo maximum likelihood
For now, the documentation is just a reference. See
`test/test-metropolis-hastings.lisp` for an example.
- [class] MC-CHAIN
A simple markov chain for Metropolis Hastings. With
temperature it is suitable for MC3.
- [accessor] TEMPERATURE MC-CHAIN (:TEMPERATURE = 1.0d0)
The PROBABILITY-RATIO of samples is raised to the
power of 1 / TEMPERATURE before calculating the acceptance
probability. This effectively flattens the peaks if TEMPERATURE >
1 which makes it easier for the chain to traverse deep valleys.
- [reader] STATE MC-CHAIN (:STATE)
This is the current sample where the chain is.
- [function] JUMP-TO-SAMPLE CHAIN JUMP &KEY (RESULT-SAMPLE (STATE CHAIN))
From the current state of CHAIN make JUMP (from the current
distribution of CHAIN) and return the sample where we landed. Reuse
RESULT-SAMPLE when possible.
- [generic-function] JUMP-TO-SAMPLE* CHAIN JUMP RESULT-SAMPLE
This function is called by JUMP-TO-SAMPLE. It is
where JUMP-TO-SAMPLE behaviour shall be customized.
- [generic-function] PREPARE-JUMP-DISTRIBUTION CHAIN
Prepare for sampling from the F(X) = Q(SAMPLE->X)
distribution. Called by RANDOM-JUMP. The around method ensures that
nothing is done unless there was a state change.
- [generic-function] RANDOM-JUMP CHAIN
Sample a jump from the current distribution of
jumps that was computed by PREPARE-JUMP-DISTRIBUTION.
- [generic-function] LOG-PROBABILITY-RATIO CHAIN SAMPLE1 SAMPLE2
Return P(SAMPLE1)/P(SAMPLE2). It's in the log
domain to avoid overflows and the ratio part is because that it may
allow computational shortcuts as opposed to calculating unnormalized
probabilities separately.
- [generic-function] LOG-PROBABILITY-RATIO-TO-JUMP-TARGET CHAIN JUMP TARGET
Return P(TARGET)/P(STATE) where JUMP is from the
current state of CHAIN to TARGET sample. This can be specialized for
speed. The default implementation just falls back on
LOG-PROBABILITY-RATIO.
- [generic-function] LOG-JUMP-PROBABILITY-RATIO CHAIN JUMP TARGET
Return Q(TARGET->STATE) / Q(STATE->TARGET) where Q
is the jump distribution and JUMP is from the current STATE of CHAIN
to TARGET sample.
- [generic-function] ACCEPTANCE-PROBABILITY CHAIN JUMP CANDIDATE
Calculate the acceptance probability of CANDIDATE
to which JUMP leads from the current STATE of CHAIN.
- [generic-function] ACCEPT-JUMP CHAIN JUMP CANDIDATE
Called when CHAIN accepts JUMP to CANDIDATE.
- [generic-function] REJECT-JUMP CHAIN JUMP CANDIDATE
Called when CHAIN rejects JUMP to CANDIDATE. It
does nothing by default, it's just a convenience for debugging.
- [generic-function] MAYBE-JUMP CHAIN JUMP CANDIDATE ACCEPTANCE-PROBABILITY
Randomly accept or reject JUMP to CANDIDATE from
the current state of CHAIN with ACCEPTANCE-PROBABILITY.
- [generic-function] JUMP CHAIN
Take a step on the markov chain. Return a boolean
indicating whether the proposed jump was accepted.
- [class] MC3-CHAIN MC-CHAIN
High probability island separated by low valley
make the chain poorly mixing. MC3-CHAIN has a number of HOT-CHAINS
that have state probabilities similar to that of the main chain but
less jagged. Often it suffices to set the temperatures of the
HOT-CHAINS higher use the very same base probability
distribution.
- [generic-function] ACCEPT-SWAP-CHAIN-STATES MC3 CHAIN1 CHAIN2
Swap the states of CHAIN1 and CHAIN2 of MC3.
- [generic-function] REJECT-SWAP-CHAIN-STATES MC3 CHAIN1 CHAIN2
Called when the swap of states of CHAIN1 and CHAIN2
is rejected. It does nothing by default, it's just a convenience for
debugging.
- [generic-function] MAYBE-SWAP-CHAIN-STATES MC3 CHAIN1 CHAIN2 ACCEPTANCE-PROBABILITY
Swap of states of CHAIN1 and CHAIN2 of MC3 with
ACCEPTANCE-PROBABILITY.
- [generic-function] JUMP-BETWEEN-CHAINS MC3
Choose two chains randomly and swap their states
with MC3 acceptance probability.
- [class] ENUMERATING-CHAIN MC-CHAIN
A simple abstract chain subclass that explicitly
enumerates the probabilities of the distribution.
- [class] TRACING-CHAIN
Mix this in with your chain to have it print trace
of acceptances/rejections.
## Game Theory
###### \[in package MICMAC.GAME-THEORY\]
- [function] FIND-NASH-EQUILIBRIUM PAYOFF &KEY (N-ITERATIONS 100)
Find a Nash equilibrium of a zero-sum game represented by PAYOFF
matrix (a 2d matrix or a nested list). PAYOFF is from the point of
view of the row player: the player who choses column wants to
minimize, the row player wants to maximize. The first value returned
is a vector of unnormalized probabilities assigned to each action of
the row player, the second value is the same for the column player
and the third is the expected payoff of the row player in the nash
equilibrium represented by the oddment vectors.
* * *
###### \[generated by [MGL-PAX](https://github.com/melisgl/mgl-pax)\]

6.1.2 Functions

Alpha-beta pruning for two player, zero-sum maximax (like minimax
but both players maximize and the score is negated when passed
between depths). Return the score of the game STATE from the point
of view of the player to move at DEPTH and as the second value the
list of actions of the principal variant.

CALL-WITH-ACTION is a function of (STATE DEPTH ACTION FN). It
carries out ACTION (returned by LIST-ACTIONS or NIL) to get the
state corresponding to DEPTH and calls FN with that state. It may
destructively modify STATE provided it undoes the damage after FN
returns. CALL-WITH-ACTION is called with NIL as ACTION for the root
of the tree, in this case STATE need not be changed. FN returns the
same kinds of values as ALPHA-BETA. They may be useful for logging.

MAYBE-EVALUATE-STATE is a function of (STATE DEPTH). If STATE at
DEPTH is a terminal node then it returns the score from the point of
view of the player to move and as the second value a list of actions
that lead from STATE to the position that was evaluated. The list of
actions is typically empty. If we are not at a terminal node then
MAYBE-EVALUATE-STATE returns NIL.

LIST-ACTIONS is a function of (STATE DEPTH) and returns a non-empty
list of legal candidate moves for non-terminal nodes. Actions are
tried in the order LIST-ACTIONS returns them: stronger moves

CALL-WITH-ACTION, MAYBE-EVALUATE-STATE and LIST-ACTIONS are
mandatory.

RECORD-BEST, if non-NIL, is a function of (DEPTH SCORE ACTIONS). It
is called when at DEPTH a new best action is found. ACTIONS is a
list of all the actions in the principle variant corresonding to the
newly found best score. RECORD-BEST is useful for graceful
degradation in case of timeout.

ALPHA and BETA are typically NIL (equivalent to -infinity,
+infinity) but any real number is allowed if the range of scores can
be boxed.

In a graph, search for nodes that with the best scores with [beam
search](http://en.wikipedia.org/wiki/Beam_search). That is, starting
from START-NODES perform a breadth-first search but at each depth
only keep BEAM-WIDTH number of nodes with the best scores. Keep the
best N-SOLUTIONS (at most) complete solutions. Discard nodes known
to be unable to get into the best N-SOLUTIONS (due to
UPPER-BOUND-FN). Finally, return the solutions and the active
nodes (the _beam_) as adjustable arrays sorted by score in
descending order.

START-NODES (a sequence of elements of arbitrary type). SCORE-FN,
UPPER-BOUND-FN, SOLUTIONP-FN, FINISHEDP-FN are all functions of one
argument: the node. SOLUTIONP-FN checks whether a node represents a
complete solution (i.e. some goal is reached). SCORE-FN returns a
real number that’s to be maximized, it’s only called for node for
which SOLUTIONP-FN returned true. UPPER-BOUND-FN (if not NIL)
returns a real number that equal or greater than the score of all
solutions reachable from that node. FINISHEDP-FN returns true iff
there is nowhere to go from the node.

EXPAND-NODE-FN is also a function of a single node argument. It
returns a sequence of nodes to ’one step away’ from its argument
node. EXPAND-BEAM-FN is similar, but it takes a vector of nodes and
returns all nodes one step away from any of them. It’s enough
provide either EXPAND-NODE-FN or EXPAND-BEAM-FN. The purpose of
EXPAND-BEAM-FN. is to allow more efficient, batch-like operations.

Find a Nash equilibrium of a zero-sum game represented by PAYOFF
matrix (a 2d matrix or a nested list). PAYOFF is from the point of
view of the row player: the player who choses column wants to
minimize, the row player wants to maximize. The first value returned
is a vector of unnormalized probabilities assigned to each action of
the row player, the second value is the same for the column player
and the third is the expected payoff of the row player in the nash
equilibrium represented by the oddment vectors.

This is very much like BEAM-SEARCH except it solves a number of
instances of the same search problem starting from different sets of
nodes. The sole purpose of PARALLEL-BEAM-SEARCH is to amortize the
cost EXPAND-BEAM-FN if possible.

EXPAND-BEAMS-FN is called with sequence of beams (i.e. it’s a
sequence of sequence of nodes) and it must return another sequence
of sequences of nodes. Each element of the returned sequence is the
reachable nodes of the nodes in the corresponding element of its
argument sequence.

PARALLEL-BEAM-SEARCH returns a sequence of solutions sequences, and
a sequence of active node sequences.

Starting from the ROOT node search the tree expanding it one node
for each playout. Finally return the mutated ROOT. ROOT may be the
root node of any tree, need not be a single node with no edges.
FRESH-ROOT-STATE is a function that returns a fresh state
corresponding to ROOT. This state will be destroyed unless special
care is taken in STATE.

Return P(SAMPLE1)/P(SAMPLE2). It’s in the log
domain to avoid overflows and the ratio part is because that it may
allow computational shortcuts as opposed to calculating unnormalized
probabilities separately.

Create a node representing the state of that EDGE
leads to from PARENT. Specialize this if you want to keep track of
the state which is not done by default as it can be expensive,
especially in light of TAKE-ACTION mutating it. The default
implementation simply creates an instance of the class of PARENT so
that one can start from a subclass of UCT-NODE and be sure that that
class is going to be used for nodes below it.

Compute the reward for a node in the tree from
OUTCOME that is the result of a playout. This is called by the
default implementation of UPDATE-UCT-STATISTICS. This is where one
typically negates depending on the parity of DEPTH in two player
games.

Play a random game from NODE with STATE and return
the outcome that’s fed into UPDATE-UCT-STATISTICS. The way the
random game is played is referred to as ‘default policy’ and that’s
what makes or breaks UCT search. This function must be
customized.

Choose an action to take from a state, in other
words an edge to follow from NODE in the tree. The default
implementation chooses randomly from the yet unvisited edges or if
there is none moves down the edge with the maximum EDGE-SCORE. If
you are thinking of customizing this, for example to make it choose
the minimum at odd depths, the you may want to consider specializing
REWARD or UPDATE-UCT-STATISTICS instead.

Return the state that corresponds to NODE. This is
not a straightforward accessor unless NODE is customized to store
it. The rest of the parameters are provided so that one can
reconstruct the state by taking the action of EDGE in the
PARENT-STATE of PARENT. It’s okay to destroy PARENT-STATE in the
process as long as it’s not stored elsewhere. This function must be
customized.

The PROBABILITY-RATIO of samples is raised to the
power of 1 / TEMPERATURE before calculating the acceptance
probability. This effectively flattens the peaks if TEMPERATURE >
1 which makes it easier for the chain to traverse deep valleys.

Increment the number of visits and update the
average reward in nodes and edges of PATH. By default, edges simply
get their visit counter incremented while nodes also get an update
to AVERAGE-REWARD based on what OUTCOME->REWARD returns.

The PROBABILITY-RATIO of samples is raised to the
power of 1 / TEMPERATURE before calculating the acceptance
probability. This effectively flattens the peaks if TEMPERATURE >
1 which makes it easier for the chain to traverse deep valleys.

High probability island separated by low valley
make the chain poorly mixing. MC3-CHAIN has a number of HOT-CHAINS
that have state probabilities similar to that of the main chain but
less jagged. Often it suffices to set the temperatures of the
HOT-CHAINS higher use the very same base probability
distribution.

An edge in the UCT tree. Represents an action taken
from a state. The value of an action is the value of its target
state which is not quite as generic as it could be; feel free to
specialize AVERAGE-REWARD for the edges if that’s not the case.

A node in the UCT tree. Roughly translates to a
state in the search space. Note that the state itself is not stored
explicity, but it can be recovered by ‘replaying’ the actions from
the starting state or by customizing MAKE-UCT-NODE.

Choose an element randomly from the [START,END) subsequence of SEQ
with given probabilities. KEY returns the unormalized probability of
an element, SUM is the sum of these unnormalized probalities
contains unnormalized probabilties. Return the element chosen and
its index.